This proposal is directed toward improving tomographic imaging in diagnostic radiology and nuclear medicine. It is predicated on the claim that significant advances will be achieved in the fidelity of the images that are reconstructed from the raw detector measurements of the tomographic scanner by changing the basic elements (called "basis functions") with which the image is built in the computer. The conventional basic elements for computerized tomographic imaging are the voxel basis functions, and the sinusoidal basis functions of Fourier analysis. Two classes of promising new basis functions have been developed: functions that are localized in space (as are the voxel basis functions), and functions that are not localized (similar in many respects to sinusoids). The new classes of basis functions are well-suited to constructing faithful digital image representations of the biological structures that have influenced the raw tomographic scanner data. The new localized basis functions have a number of very desirable properties not shared by voxels: they are rotationally symmetric, their Fourier transforms are effectively localized, and they have continuous derivatives of any desired order. The new non-localized basis functions are designed to perform a spatially-variant filtering operation that is required by a non-iterative method of 3D image reconstruction developed by the Principal Investigator. The specific aims are to develop mathematical theory, efficient computer algorithms, application-specific implementations and evaluation criteria for (1) methods of iterative reconstruction from projections, (2) methods of estimating the fundamental limits on the performance of the reconstruction process, and (3) methods of non-iterative 3D reconstruction from projections. For specified imaging tasks, the level of statistical significance will be found for rejection of the null hypothesis that two methods perform a task equally well, in favor of the alternative hypothesis that one method performs the task better. The basis functions of the image representation are the essential core of all methods for computerized image reconstruction, irrespective of the medical imaging modality (e.g., CT, PET, SPECT, MRI). The development of new computer algorithms and their associated image representations will enable the full potential of scanners for functional imaging in emission tomography (PET and SPECT) to be realized by extracting as much information as possible from fully-3D low-statistics projection data.